Hexagonal tiling honeycomb

Hexagonal tiling honeycomb
Perspective projection view within Poincaré disk model
Type	Hyperbolic regular honeycomb Paracompact uniform honeycomb
Schläfli symbols	{6,3,3} t{3,6,3} 2t{6,3,6} 2t{6,3[3]} t{3[3,3]}
Coxeter diagrams	↔ ↔ ↔ ↔
Cells	{6,3}
Faces	hexagon {6}
Edge figure	triangle {3}
Vertex figure	tetrahedron {3,3}
Dual	Order-6 tetrahedral honeycomb
Coxeter groups	${\displaystyle {\overline {V}}_{3}}$, [3,3,6] ${\displaystyle {\overline {Y}}_{3}}$, [3,6,3] ${\displaystyle {\overline {Z}}_{3}}$, [6,3,6] ${\displaystyle {\overline {VP}}_{3}}$, [6,3[3]] ${\displaystyle {\overline {PP}}_{3}}$, [3[3,3]]
Properties	Regular

In the field of hyperbolic geometry, the hexagonal tiling honeycomb is one of 11 regular paracompact honeycombs in 3-dimensional hyperbolic space. It is paracompact because it has cells composed of an infinite number of faces. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity.

The Schläfli symbol of the hexagonal tiling honeycomb is {6,3,3}. Since that of the hexagonal tiling is {6,3}, this honeycomb has three such hexagonal tilings meeting at each edge. Since the Schläfli symbol of the tetrahedron is {3,3}, the vertex figure of this honeycomb is a tetrahedron. Thus, four hexagonal tilings meet at each vertex of this honeycomb, six hexagons meet at each vertex, and four edges meet at each vertex.^[1]